Derivative of \( \displaystyle - \sqrt{x^{2} - 2 x + 2} \)

Problem 2.1831 · hard

1. \[ \frac{d}{d x} \left(- \sqrt{x^{2} - 2 x + 2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} \sqrt{x^{2} - 2 x + 2} \]
   constant-multiple rewritePull out the constant -1. Rewrite the square root using the exponential and logarithm.✓ Proved
3. \[ = - \sqrt{x^{2} - 2 x + 2} \frac{d}{d x} \frac{\ln{\left(x^{2} - 2 x + 2 \right)}}{2} \]
   chainApply the chain rule for the exponential function.✓ Proved
4. \[ = - \frac{\sqrt{x^{2} - 2 x + 2} \frac{d}{d x} \ln{\left(x^{2} - 2 x + 2 \right)}}{2} \]
   constant-multiplePull out the constant 1/2.✓ Proved
5. \[ = - \frac{\frac{d}{d x} \left(x^{2} - 2 x + 2\right)}{2 \sqrt{x^{2} - 2 x + 2}} \]
   chainApply the chain rule for the logarithm.✓ Proved
6. \[ = - \frac{2 x - 2}{2 \sqrt{x^{2} - 2 x + 2}} \]
   derivative algebraDifferentiate the inner polynomial. Substitute the exponential expression back to its square root form.✓ Proved
7. \[ = \frac{2 - 2 x}{2 \sqrt{x^{2} - 2 x + 2}} \]
   algebraCombine the terms into a single fraction.✓ Proved
8. \[ = \frac{1 - x}{\sqrt{x^{2} - 2 x + 2}} \]
   simplifySimplify the fraction by canceling the common factor of 2.✓ Proved